Let $J = \left\{k \in \{1, 2, \ldots, n\}, \left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0\right\}$ be the set of indices $k$ such that $\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0$. (a) Show that $J \neq \varnothing$. (b) Let $\ell \notin J$. Show that $\lambda_\ell$ is an eigenvalue of $B$. (c) Suppose that $J = \{j\}$ for some $j \in \{1, 2, \ldots, n\}$. Show that the eigenvalues of $B$ are $$\left(\lambda_1, \lambda_2, \ldots, \lambda_{j-1}, \lambda_j + 1, \lambda_{j+1}, \ldots, \lambda_n\right).$$
Let $J = \left\{k \in \{1, 2, \ldots, n\}, \left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0\right\}$ be the set of indices $k$ such that $\left\langle \mathbf{w}_k, \mathbf{u} \right\rangle \neq 0$.\\
(a) Show that $J \neq \varnothing$.\\
(b) Let $\ell \notin J$. Show that $\lambda_\ell$ is an eigenvalue of $B$.\\
(c) Suppose that $J = \{j\}$ for some $j \in \{1, 2, \ldots, n\}$. Show that the eigenvalues of $B$ are
$$\left(\lambda_1, \lambda_2, \ldots, \lambda_{j-1}, \lambda_j + 1, \lambda_{j+1}, \ldots, \lambda_n\right).$$